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Resonant Weighted Nonlocal Schrödinger Equation with Gauge Invariance, Conservation Laws and Measurable Phase Detuning

L. Yildiz, D. Kayki, E. Gudekli

Abstract

We present a gauge-invariant Schrödinger-type evolution that combines (i) weighted local diffusion, (ii) symmetric nonlocal exchange through a kernel operator, and (iii) a mean-free phase-resonant drive. The resulting Resonant Weighted Nonlocal Schrödinger (RWNS) equation exactly conserves mass and, when the drive is absent, admits a Hamiltonian structure with energy conservation. Under standard assumptions on the weight, kernel, and nonlinearity, we establish local well-posedness in $H^1$ and provide defocusing conditions for global continuation. Linearization yields a dispersion relation in which the nonlocal kernel and the mean-free phase field contribute additively to a measurable spectral detuning. Building on this, we define two observables: a wavenumber-resolved detuning $Δω(k)$ and a kernel-contrast functional $Ξ[ψ]$ that isolates the nonlocal exchange. We outline feasible implementations in nonlinear-optical lattices and cavity-assisted cold-atom platforms, and discuss conceptual links to propagation-induced phase signatures in astrophysical media. The RWNS model thus offers a compact and analytically tractable framework that unifies weighted local dynamics, symmetric nonlocality, and a mean-free phase drive, yielding clear, testable predictions for laboratory measurements and, in principle, precision timing data.

Resonant Weighted Nonlocal Schrödinger Equation with Gauge Invariance, Conservation Laws and Measurable Phase Detuning

Abstract

We present a gauge-invariant Schrödinger-type evolution that combines (i) weighted local diffusion, (ii) symmetric nonlocal exchange through a kernel operator, and (iii) a mean-free phase-resonant drive. The resulting Resonant Weighted Nonlocal Schrödinger (RWNS) equation exactly conserves mass and, when the drive is absent, admits a Hamiltonian structure with energy conservation. Under standard assumptions on the weight, kernel, and nonlinearity, we establish local well-posedness in and provide defocusing conditions for global continuation. Linearization yields a dispersion relation in which the nonlocal kernel and the mean-free phase field contribute additively to a measurable spectral detuning. Building on this, we define two observables: a wavenumber-resolved detuning and a kernel-contrast functional that isolates the nonlocal exchange. We outline feasible implementations in nonlinear-optical lattices and cavity-assisted cold-atom platforms, and discuss conceptual links to propagation-induced phase signatures in astrophysical media. The RWNS model thus offers a compact and analytically tractable framework that unifies weighted local dynamics, symmetric nonlocality, and a mean-free phase drive, yielding clear, testable predictions for laboratory measurements and, in principle, precision timing data.
Paper Structure (31 sections, 42 equations, 5 figures)

This paper contains 31 sections, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Equation and assumptions
  3. Standing hypotheses
  4. Mass conservation
  5. Energy functional and conservation for $\gamma=0$
  6. Energy functional and conservation
  7. Energy functional.
  8. Geometric structure: symplectic form, momentum map, and Noether symmetries
  9. Nonlocal quadratic form and first variation.
  10. Hamiltonian form of the RWNS flow ($\gamma=0$).
  11. Global $U(1)$ action and momentum map.
  12. Weighted geometric interpretation (local part).
  13. Translation symmetry and conserved momentum (homogeneous case).
  14. Summary.
  15. Energy conservation (statement and short proof).
  16. ...and 16 more sections

Figures (5)

  • Figure 1: Full dispersion vs. Euclidean baseline. Red: $\omega_0(k)$; blue dashed: $w_0|k|^2+U_0$; green shading: residual $\Delta\omega(k)=\omega_0(k)-(w_0|k|^2+U_0)$, cf. \ref{['eq:dispersion']}.
  • Figure 2: Dispersion residual (O1). Blue symbols: measured $\Delta\omega(k)$; red line: RWNS fit $\kappa(\widehat{\mathcal{K}}(0)-\widehat{\mathcal{K}}(k))$; green dashed: small-$|k|$ quadratic trend, cf. \ref{['eq:dispersion']}.
  • Figure 3: Small-$|k|$ slope diagnostic for O1. The ratio $\Delta\omega(k)/|k|^2$ versus $|k|$ approaches the predicted plateau $\kappa\sigma^2/2$ (red). Blue markers: data; green dashed: noiseless model. Confirms the second-moment slope in \ref{['eq:dispersion']}.
  • Figure 4: Kernel-contrast (O2), cf. \ref{['eq:xi-def']}. Blue band: interquartile range over snapshots; green points: representative snapshots; red line: prediction from O1 parameters. O2 is drive-agnostic and validates the kernel independently of the phase drive \ref{['eq:centered-mean']}.
  • Figure 5: Phase-drive sideband ratio (O3), cf. \ref{['eq:ratio']}. Blue symbols: measured $R(q,k)$; red line: fit including linewidth $\eta$; green dashed: $\eta\!\to\!0$ reference. Kernel parameters $(\kappa,\widehat{\mathcal{K}})$ are fixed by O1; the fit returns $|\gamma\,\Phi_q|$.